In this paper we consider a class of {\it conditional McKean-Vlasov SDEs} (CMVSDE for short). Such an SDE can be considered as extended version SDEs with common noises, well the general so-called mean-field (CMFSDE) studied previously by authors [1, 14], but some fundamental differences. particular, due to lack compactness iterated laws, existing arguments Schauder's fixed point theorem do not ...

We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and deletion of edges together with their end points. Like in the case of deletion and contraction only (W. Tutte, 1954), it turns out that there is a most general polynomial satisfying such recurrence relations, which we call ξ(G, x, y, z). We show that the new p...

We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and edge extraction, i.e., deletion of edges together with their end points. Like in the case of deletion and contraction only (J.G. Oxley and D.J.A. Welsh 1979), it turns out that there is a most general polynomial satisfying such recurrence relations, which we c...

Given a formal model of the behavior of a system, an objective and some notion of control the goal of controller synthesis [RW87, Won89] is to construct a (finite-state) controller that ensures that the system always satisfies the objective. Often, the controller can base its decisions only on limited observations of the system. This notion of limited observability induces a partial-information...

A general approach for developing distribution free tests for general linear models based on simplicial depth is presented. In most relevant cases, the test statistic is a degenerated U-statistic so that the spectral decomposition of the conditional expectation of the kernel function is needed to derive the asymptotic distribution. A general formula for this conditional expectation is derived. ...

Systems need to be updated to last for a long time in a dynamic environment, and to cope with changing requirements. It is important for updates to preserve the desirable properties of the system under update, while possibly enforcing new ones. Here we consider a simple yet general update mechanism, which replaces a component of the system with a new one. The context, i.e., the rest of the syst...